Uniformly convex functions on Banach spaces
Jonathan Michael Borwein, Antonio José Guirao, Petr Hájek, Jon D. Vanderwerff · Proceedings of the American Mathematical Society · 2008
Given a Banach space ($X$,$\|\cdot \|$), we study the connection between uniformly convex functions $f:X \to \mathbb {R}$ bounded above by $\|\cdot \|^p$ and the existence of norms on $X$ with moduli of convexity of power type. In particular, we show that there exists a uniformly convex function $f:X \to \mathbb {R}$ bounded above by $\|\cdot \|^2$ if and only if $X$ admits an equivalent norm with modulus of convexity of power type 2.